Embedded newform 43.9.f.a.22.4

• Label: 43.9.f.a.22.4
• Level: $43$
• Weight: $9$
• Character: 43.22
• Analytic conductor: $17.517$
• Analytic rank: $0$
• Dimension: $174$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.5172802326\)
Analytic rank:	\(0\)
Dimension:	\(174\)
Relative dimension:	\(29\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			22.4
Character	\(\chi\)	\(=\)	43.22
Dual form			43.9.f.a.2.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.5206 + 23.9228i) q^{2} +(-29.0150 - 60.2502i) q^{3} +(-279.962 - 351.061i) q^{4} +(-438.800 + 100.153i) q^{5} +1775.62 q^{6} -3131.86i q^{7} +(4996.73 - 1140.47i) q^{8} +(1302.50 - 1633.28i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q - 7 q^{2} - 7 q^{3} + 4021 q^{4} - 7 q^{5} + 1780 q^{6} - 5383 q^{8} + 56490 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{14}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.5206	+	23.9228i	−0.720038	+	1.49517i	0.142832	+	0.989747i	\(0.454379\pi\)
							−0.862869	+	0.505427i	\(0.831335\pi\)
\(3\)	−29.0150	−	60.2502i	−0.358210	−	0.743830i	0.641519	−	0.767107i	\(-0.278304\pi\)
							−0.999729	+	0.0232769i	\(0.992590\pi\)
\(5\)	−438.800	+	100.153i	−0.702079	+	0.160245i	−0.558635	−	0.829413i	\(-0.688675\pi\)
							−0.143444	+	0.989658i	\(0.545818\pi\)
\(7\)		−	3131.86i		−	1.30440i	−0.758047	−	0.652200i	\(-0.773846\pi\)
							0.758047	−	0.652200i	\(-0.226154\pi\)
\(11\)	−6444.85	+	8081.59i	−0.440192	+	0.551983i	−0.951594	−	0.307358i	\(-0.900555\pi\)
							0.511402	+	0.859342i	\(0.329126\pi\)
\(13\)	3311.51	+	14508.7i	0.115945	+	0.507989i	0.999233	+	0.0391580i	\(0.0124676\pi\)
							−0.883288	+	0.468831i	\(0.844675\pi\)
\(17\)	−3312.06	+	14511.1i	−0.0396555	+	0.173742i	−0.990878	−	0.134765i	\(-0.956972\pi\)
							0.951222	+	0.308507i	\(0.0998292\pi\)
\(19\)	−30059.5	+	23971.7i	−0.230657	+	0.183943i	−0.731998	−	0.681307i	\(-0.761412\pi\)
							0.501340	+	0.865250i	\(0.332840\pi\)
\(23\)	69185.0	−	86755.3i	0.247230	−	0.310016i	−0.642696	−	0.766121i	\(-0.722184\pi\)
							0.889926	+	0.456105i	\(0.150756\pi\)
\(29\)	−92334.2	+	191734.i	−0.130548	+	0.271086i	−0.955989	−	0.293403i	\(-0.905212\pi\)
							0.825441	+	0.564489i	\(0.190927\pi\)
\(31\)	−114319.	−	55053.3i	−0.123786	−	0.0596124i	0.370966	−	0.928647i	\(-0.379027\pi\)
							−0.494752	+	0.869034i	\(0.664741\pi\)
\(37\)		−	655296.i		−	0.349648i	−0.984600	−	0.174824i	\(-0.944064\pi\)
							0.984600	−	0.174824i	\(-0.0559356\pi\)
\(41\)	2.24206e6	+	1.07972e6i	0.793435	+	0.382098i	0.786275	−	0.617876i	\(-0.212007\pi\)
							0.00715946	+	0.999974i	\(0.497721\pi\)
\(43\)	1.04462e6	+	3.25530e6i	0.305551	+	0.952176i
							\(47\)	5.66056e6	+	7.09812e6i	1.16003	+	1.45463i	0.866849	+	0.498571i	\(0.166142\pi\)
0.293178	+	0.956058i	\(0.405287\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.f.a.22.4	yes	174
43.2	odd	14	inner	43.9.f.a.2.4	✓	174

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.f.a.2.4	✓	174	43.2	odd	14	inner
43.9.f.a.22.4	yes	174	1.1	even	1	trivial